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Workshop Programme

for period 12 - 16 December 2005

Einstein Constraint Equations

12 - 16 December 2005


Monday 12 December
08:30-09:55 Registration
10:00-11:00 Bartnik, R (Monash)
  Hilbert structure on the ADM phase space Sem 1

The Einstein equations can be formulated as a densely defined flow on a phase space modelled on the Hilbert space H2 x H1 with appropriate decay conditions. I will show that the constraint system determines a smooth Hilbert submanifold and the ADM energy-momentum extends smoothly to the entire phase space. These constructions are motivated by the variational definition of quasi-local mass.

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11:00-11:30 Coffee
11:30-12:30 Huisken, G (Max Planck Institute for Gravitation)
  Radial foliations of asymptotically flat 3-manifolds Sem 1
12:30-13:30 Lunch at Wolfson Court
14:30-15:30 Dain, S (Max Planck Institute)
  Spin-mass inequality for axisymmetric black holes Sem 1

Abstract: In this talk I will discuss the physical relevance of the inequality J < m^2, where m and J are the total mass and angular momentum, for axially symmetric (non-stationary) black holes. In particular I will show that for any vacuum, maximal, complete, asymptotically flat, axisymmetric initial data close to extreme Kerr data, this inequality is satisfied. The proof consists in showing that extreme Kerr is a local minimum of the mass.

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15:30-16:00 Tea
16:00-16:30 Delay, E (Universite d'Avignon)
  Nonsingular stationary metrics with a negative cosmological constant Sem 1

In a joint work with Piotr Chrusciel, we construct infinite dimensional families of non-singular stationary space times, solutions of the vacuum Einstein equations with a negative cosmological constant.

16:30-18:00 Wine Reception
18:30-19:45 Dinner at Wolfson Court (Residents only)
Tuesday 13 December
10:00-11:00 Khuri, M (Stanford)
  Global bounds and new existence theorems for the Yamabe problem Sem 1
11:00-11:30 Coffee
11:30-12:30 Bahri, A (Rutgers)
  A variational approach to the Yamabe problem Sem 1
12:30-13:30 Lunch at Wolfson Court
14:30-15:30 Pacard, F (Paris 12)
  Singular solutions of the Yamabe equation Sem 1

The existence of conformal metrics with constant (positive) scalar curvature on subdomains of the sphere is related to the existence of singular solutions for some semilinear elliptic equation.

I will review the sufficient conditions which are known to ensure the existence of singular solutions for this equation.

15:30-16:00 Tea
16:00-16:30 Pollack, D (Washington)
  Constructing solutions of the constraint equations with sources: the Einstein-Scalar field system Sem 1

We present recent work (with J. Isenberg and Y. Choquet-Bruhat) concerning the construction of solutions of the Einstein-Scalar field constraint equations via the conformal method.

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18:30-19:45 Dinner at Wolfson Court (Residents only)
Wednesday 14 December
10:00-11:00 Brendle, S (Princeton)
  Global convergence of the Yamabe flow Sem 1
11:00-11:30 Coffee
11:30-12:30 Maxwell, D (Alaska Fairbanks)
  Rough initial data Sem 1

The story of constant mean curvature $H^s$ solutions of the constraint equations with $s>3/2$ has largely been completed, both for asymptotically Euclidean and compact manifolds. It turns out that the standard existence results for smooth solutions extend fully and naturally to the low regularity setting. In this talk I will describe how these results were obtained. One point of interest, even for smooth solutions, is that the rough theory leads to a unified and simpler approach for working with the various cases of the CMC conformal method on compact manifolds.

12:30-13:30 Lunch at Wolfson Court
18:30-19:45 Dinner at Wolfson Court (Residents only)
Thursday 15 December
10:00-11:00 Pfeiffer, H (Caltech)
  Numerical construction of the solutions of the constraint equations Sem 1
11:00-11:30 Coffee
11:30-12:30 Lindblom, L (Caltech)
  Optimal constraint projection in general relativity Sem 1
12:30-13:30 Lunch at Wolfson Court
14:30-15:30 Alexakis, S (Princeton)
  Global conformal invariants and their applications Sem 1

We discuss our recent partial confirmation of a conjecture of Deser and Schwimmer regarding the structure of "global conformal invariants". These are scalar quantites whose integrals over compacr manifolds remain invariant under conformal changes of the underlying metric. We also discuss the implications that the full conjecture would have regarding the notions of Q-curvature, and of the renormalized volume and conformal anomalies of conformally compact Einstein manifolds

15:30-16:00 Tea
16:00-16:30 Corvino, J (Lafayette College)
  Some applications of scalar curvature deformation in general relativity Sem 1

The past several years have seen much activity in constructing solutions of the constraint equations by using geometric gluing techniques. These results require an understanding of the scalar curvature operator (and more generally the constraint operator), from the conformal as well as the underdetermined-elliptic points of view. We discuss several applications of these techniques, including the existence of asymptotically simple vacuum spacetimes, and a construction of multi-horizon initial data with trivial topology.

19:30-18:00 Conference Dinner at Clare College (Bar open 18:45 - 23:30)
Friday 16 December
10:00-11:00 Galloway, GJ (Miami)
  Rigidity and positivity of mass for asymptotically hyperbolic manifolds Sem 1

We discuss an approach to the proof of positivity of mass without spin assumption, for asymptotically hyperbolic Riemannian manifolds, based on the general methodology of Schoen and Yau. Our approach makes use of the "BPS brane action" introduced by Witten and Yau in their work on the AdS/CFT correspondence, and takes hints from work of Lohkamp. This is joint work with Lars Andersson and Mingliang Cai.

11:00-11:30 Coffee
11:30-12:30 Miao, P (Santa Barbara)
  On problems related to Bartnik's definition of quasi-local mass (sponsored by CQG) Sem 1
12:30-13:30 Lunch at Wolfson Court
14:30-15:00 Maerten, D (Montpellier II)
  Positive energy theorem for asymptotically hyperbolic manifolds Sem 1

General Relativity is a geometrical theory of gravity which asserts that the geometry of space-time is closely related to matter. There exists some consistent definition for total energy (and momentum) of isolated systems which by definition are manifolds whose metric approaches a background metric (Euclidean or hyperbolic). The positive mass theorem can be considered as attempts at understanding the relationship between the local energy density (namely the stress-energy tensor) and the total energy of a space-time. On one hand P. T. Chrusciel and G. Nagy rigorously defined in a recent work notions of mass and momentum for manifolds which are asymptotic to a standard hyperbolic slice of Minkowski space-time. On the other hand P. T. Chrusciel and M. Herzlich proved a positive mass theorem for Riemannian asymptotically hyperbolic manifolds. My work extends this result for orientable 3-dimensional manifolds which are asymptotic to a standard hyperbolic slice of anti-de Sitter space-time in the following way: we define a sesquilinear form Q which is closely related to the energy-momentum and prove, under the relevant energy condition, that Q is a nonnegative Hermitian form which is in fact definite unless our manifold is isometrically embeddable in anti-de Sitter.

15:00-15:30 O'Murchadha, N (University College, Cork)
  Why we should not take the Liu-Yau quasi-local mass seriously Sem 1

The Liu-Yau mass is a true mass, it is frame independent. However, the Liu-Yau mass is bigger than the Brown-York energy on any surface for which both can be defined. Further, if I take a sequence of `coordinate spheres' on any spacelike slice, both the Liu-Yau mass and the Brown-York energy asymptote to the ADM mass (which is really an energy, the `0'-th component of a Lorentz covariant 4-vector). This means that in any asymptotically flat spacetime, I can find a 2-surface with unboundedly large Liu-Yau mass. This is even true for Minkowski space.

15:30-16:00 Tea
18:30-19:45 Dinner at Wolfson Court (Residents only)

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